Weingarten equations
Weingarten equations is about derivative of the unit normal vector \( \vec{N} \).
These equations were established in 1861 by German mathematician Julius Weingarten.
The Weingarten equation is
\( {H^2}\vec{N}_1=( FM-GL )\vec{r}_1+( FL-EM )\vec{r}_2\) and
\( {H^2}\vec{N}_2=( FN-GM )\vec{r}_1+( FM-EN )\vec{r}_2\)
The matrix equivalent of the Weingarten equation is
\( {H^2} \begin{pmatrix} \vec{N}_1 \\ \vec{N}_2 \end{pmatrix}=\begin{pmatrix} FM-GL & FL-EM \\ FN-GM & FM-EN \end{pmatrix} \begin{pmatrix} \vec{r}_1 \\ \vec{r}_2 \end{pmatrix} \)
In a surface, show that
- \( {H^2}\vec{N}_1=( FM-GL )\vec{r}_1+( FL-EM )\vec{r}_2\) and
- \( {H^2}\vec{N}_2=( FN-GM )\vec{r}_1+( FM-EN )\vec{r}_2\)
Also, verify that
\( H\vec{N}_1\times \vec{N}_2=( LN-{M^2} )\vec{N}\)
Proof
Let \( S:\vec{r}=\vec{r}( u,v )\) be a surface and \( \vec{N}\) be unit normal.
Then,
\( d\vec{N}\) lies in tangent plane of S.
Thus, we write
\( \vec{N}_1=a\vec{r}_1+b\vec{r}_2\) (A)
\( \vec{N}_2=c\vec{r}_1+d\vec{r}_2\) (B)
Taking dot product on both sides of (A) by \( \vec{r}_1\) we get
\( \vec{N}_1.\vec{r}_1=a\vec{r}_1^2+b\vec{r}_1.\vec{r}_2\)
or
\( -L = aE+bF \)
or
\( aE+b F + L =0 \) (1)
Again,
Taking dot product on both sides of (A) by \( \vec{r}_2\) we get
\( \vec{N}_1.\vec{r}_2=a\vec{r}_1.\vec{r}_2+b\vec{r}_2^2\)
or
\( -M = a F +b G \)
or
\( aF+bG+M=0 \) (2)
Now,
The cross-multiplication method is applied.
Solving, equations (1) and (2), for \( a \) and \( b \) ,we get
\( \frac{a}{ FM-GL } = \frac{b}{ FL-EM }= \frac{1}{EG-F^2}\)
or
\( \frac{a}{ FM-GL } = \frac{b}{ FL-EM }= \frac{1}{H^2}\)
Equating first and third identity, similarly equating second and third identity, we get
or
\( a=\frac{ FM-GL }{ H^2} , b= \frac{ FL-EM }{ H^2}\)
Substituting, value of \( a \) and \( b\) in (A) we get
\( \vec{N}_1=a\vec{r}_1+b\vec{r}_2\)
or
\( \vec{N}_1=\frac{ FM-GL }{ H^2} \vec{r}_1+\frac{ FL-EM }{ H^2} \vec{r}_2\)
or
\( {H^2}\vec{N}_1=( FM-GL )\vec{r}_1+( FL-EM )\vec{r}_2\)
The first Weingarten equation established.
Similarly,
Taking dot product on both sides of (B) by \( \vec{r}_1\) we get
\( \vec{N}_2.\vec{r}_1=c\vec{r}_1^2+d\vec{r}_1.\vec{r}_2\)
or
\( -M = cE+dF \)
or
\( cE+d F + M =0 \) (3)
Again,
Taking dot product on both sides of (B) by \( \vec{r}_2\) we get
\( \vec{N}_2.\vec{r}_2=c\vec{r}_1.\vec{r}_2+d\vec{r}_2^2\)
or
\( -N = c F +d G \)
or
\( cF+dG+N=0 \) (4)
Now,
Solving, equations (3) and (4), for \( c \) and \( d \) ,we get
The cross-multiplication method is applied.
\( \frac{c}{ FN-GM } = \frac{d}{ FM-EN }= \frac{1}{EG-F^2}\)
or
\( \frac{c}{ FN-GM } = \frac{d}{ FM-EN }= \frac{1}{H^2}\)
Equating first and third identity, similarly equating second and third identity, we get
\( c=\frac{ FN-GM }{ H^2} , d= \frac{ FM-EN }{ H^2}\)
Substituting, value of \( c \) and \(d\) in (B) we get
\( \vec{N}_2=c\vec{r}_1+d\vec{r}_2\)
or
\( \vec{N}_2=\frac{ FN-GM }{ H^2} \vec{r}_1+\frac{ FM-EN }{ H^2} \vec{r}_2\)
or
\( H^2 \vec{N}_2=( FN-GM ) \vec{r}_1+( FM-EN ) \vec{r}_2\)
The second Weingarten equation established.
Finally,
The Weingarten equation is
\( {H^2}\vec{N}_1=( FM-GL )\vec{r}_1+( FL-EM )\vec{r}_2\) and
\( {H^2}\vec{N}_2=( FN-GM )\vec{r}_1+( FM-EN )\vec{r}_2\)
Taking cross product of Weingarten equations, we get
\( {H^2}\vec{N}_1 \times {H^2}\vec{N}_2 = \left\{ ( FM-GL )\vec{r}_1 + ( FL-EM )\vec{r}_2 \right \} \times \left \{ ( FN-GM )\vec{r}_1 + ( FM-EN )\vec{r}_2 \right \} \)
or\( H^4\vec{N}_1 \times \vec{N}_2 = ( FM-GL )\vec{r}_1 \times ( FM-EN ) \vec{r}_2 + (FL-EM )\vec{r}_2 \times ( FN-GM )\vec{r}_1 \)
or\( H^4\vec{N}_1 \times \vec{N}_2 = ( FM-GL ) ( FM-EN ) (\vec{r}_1 \times \vec{r}_2 )+ (FL-EM ) ( FN-GM ) ( \vec{r}_2 \times \vec{r}_1) \)
or\( H^4\vec{N}_1 \times \vec{N}_2 = ( FM-GL ) ( FM-EN ) ( \vec{r}_1 \times \vec{r}_2 ) - (FL-EM ) ( FN-GM ) (\vec{r}_1 \times \vec{r}_2) \)
or\( H^4\vec{N}_1 \times \vec{N}_2 = ( FM-GL ) ( FM-EN ) H \vec{N} - (FL-EM ) ( FN-GM ) H \vec{N} \)
or\( H^4 \vec{N}_1 \times \vec{N}_2 = \{ ( FM-GL ) ( FM-EN ) - (FL-EM ) ( FN-GM ) \} H\vec{N} \)
or\( {H^4}\vec{N}_1 \times \vec{N}_2 = ( EG-F^2 ) ( LN-M^2 ) H \vec{N} \)
or\( {H^4}\vec{N}_1 \times \vec{N}_2 = H^2 ( LN-M^2 ) H \vec{N} \)
or\( {H^4}\vec{N}_1 \times \vec{N}_2 = H^3 ( LN-M^2 ) \vec{N} \)
or\( H\vec{N}_1 \times \vec{N}_2 = ( LN-M^2 ) \vec{N} \)
This completes.
We know that
A point \( p \) on a surface is called umbilical if and only if the principal curvatures at \( p \) are equal, i.e., \( k_1 = k_2 \). So, If \( p \) is umbilical, then \(\mu = \frac{k_1 + k_2}{2} = \frac{k + k}{2} = k\)
Next
\(K = k_1 k_2 = k \times k = k^2\)
This shows that if \( p \) is an umbilical point, then \( \mu^2 = K \)
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